Image-forming MR methods which utilize the interaction between magnetic fields and nuclear spins in order to form two-dimensional or three-dimensional images are widely used nowadays, notably in the field of medical diagnostics, because for the imaging of soft tissue they are superior to other imaging methods in many respects, do not require ionizing radiation and are usually not invasive.
According to the MR method in general, the body of the patient to be examined is arranged in a strong, uniform magnetic field whose direction at the same time defines an axis (normally the z-axis) of the co-ordinate system on which the measurement is based. The magnetic field produces different energy levels for the individual nuclear spins in dependence on the magnetic field strength which can be excited (spin resonance) by application of an electromagnetic alternating field (RF field) of defined frequency (so-called Larmor frequency, or MR frequency). From a macroscopic point of view, the distribution of the individual nuclear spins produces an overall magnetization which can be deflected out of the state of equilibrium by application of an electromagnetic pulse of appropriate frequency (RF pulse) while the magnetic field of the RF pulse extends perpendicular to the z-axis, so that the magnetization performs a precession about the z-axis. This motion of the magnetization describes a surface of a cone whose angle of aperture is referred to as flip angle. The magnitude of the flip angle is dependent on the strength and the duration of the applied electromagnetic pulse. In the case of a so-called 90° pulse, the spins are deflected from the z axis to the transverse plane (flip angle 90°). The RF pulse is radiated toward the body of the patient via a RF coil arrangement of the MR device. The RF coil arrangement typically surrounds the examination volume in which the body of the patient is placed.
After termination of the RF pulse, the magnetization relaxes back to the original state of equilibrium, in which the magnetization in the z direction is built up again with a first time constant T1 (spin lattice or longitudinal relaxation time), and the magnetization in the direction perpendicular to the z direction relaxes with a second time constant T2 (spin-spin or transverse relaxation time). The variation of the magnetization can be detected by means of receiving RF antennas or coils which are arranged and oriented within the examination volume of the MR device in such a manner that the variation of the magnetization is measured in the direction perpendicular to the z-axis. The decay of the transverse magnetization is accompanied, after application of, for example, a 90° pulse, by a transition of the nuclear spins (induced by local magnetic field inhomogeneities) from an ordered state with the same phase to a state in which all phase angles are uniformly distributed (dephasing). The dephasing can be compensated by means of a refocusing pulse (for example a 180° pulse). This produces an echo signal (spin echo) in the receiving coils.
In order to realize spatial resolution in the body, linear magnetic field gradients extending along the three main axes are superposed on the uniform magnetic field, leading to a linear spatial dependency of the spin resonance frequency. The signal picked up in the receiving coils then contains components of different frequencies which can be associated with different locations in the body. The signal data obtained via the receiving RF antennas or coils corresponds to the spatial frequency domain and is called k-space data. The k-space data usually includes multiple lines acquired with different phase encoding. Each line is digitized by collecting a number of samples. A set of k-space data is converted to a MR image by means of Fourier transformation or by other per se known reconstruction techniques.
Imaging speed is important in many MR imaging applications. However, the speed at which the MR signal data required for reconstruction of a MR image can be collected is fundamentally limited by physical and physiological constraints. Hence, many recent developments in the MR imaging field aim at reducing the amount of acquired signal data without degrading the quality of the reconstructed MR image. Among many of such developments the theory of compressed sensing (CS) has great potential for significant signal data reduction. In CS theory, a signal data set which has a sparse representation in a transform domain can be recovered from undersampled measurements by application of a suitable regularization algorithm. The possibility of undersampling leads to a significantly reduced acquisition time. As a mathematical framework for signal sampling and reconstruction, CS prescribes the conditions under which a signal data set can be reconstructed exactly or at least with high image quality even in cases in which the k-space sampling density is far below the Nyquist criterion, and it also provides the methods for such reconstruction. In most existing CS-based MR acquisition and reconstruction schemes the basic CS formulation is used which exploits only the prerequisite that the MR signal data is sparse in a transform domain. For example, M. Lustig et al. have proposed the application of CS for rapid MR imaging (M. Lustig et al.: “Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging”, Magnetic Resonance in Medicine, 58, 1182-1195, 2007). It is also known that, since additional prior information about the unknown MR image may be available in certain applications, it is advantageous to incorporate this prior information into the CS reconstruction.
As already mentioned above, scan time is a critical factor in MR imaging. For this reason, the temporal or spatial resolution of the acquired MR images as well as the number of scans within a clinical examination is often limited. Due to time limitations in clinical practice, MR imaging scans requiring a particularly long scan time are sometimes even completely excluded from the protocol or the signal data has to be acquired at low resolution. The above mentioned CS technique has successfully helped to reduce the scan time required for reconstruction of a single MR image. However, the time required for a complete MR examination which includes the acquisition and reconstruction of several MR images of different contrast types for obtaining the desired diagnostic information, still exceeds the time limits in many practical cases.